atmospheric refraction error and its compensation for ...atmospheric refraction error and its...
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rethnicai RepOrt686
Atmospheric Refraction Error and ItsCompensation for Passive, Optical Sensors
K-P. Dunn
4 June 1984
Lincoln LaboratoryMASSA(II 'Sil'TTS INSTITUTE OF TECHNOLOGY
LE.V\ 'iO\. 11 4.SS'q4(111'sE'.sET~
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
LINCOLN LABORATORY
ATMOSPHERIC REFRACTION ERROR AND ITS
COMPENSATION FOR PASSIVE OPTICAL SENSORS
K.l'. DUNN
Group 32
TECH N I(A L R EPOTIT 686
4 JUNE 1984
Accession For
NTIS GRA&IDTIC TAB F]Unannounced-Justificatio Approved for public rleawe ulitrilution unlimileti.
By'Distribution/
Availability C odes STTSAvail and/or a W ELECE
Dist Special AUG 8
D
LEXINGTON MASSACHUSETTS
ABSTRACT
A wide range of passive optical sensor applications
often require the sensor to operate within the atmosphere
while the objects it examines are outside the atmosphere. The
refraction of light by the earth's atmosphere becomes a
significant error source when the objects are at low elevation
angle. When the measurement accuracy at low elevation angle
is important for the sensor application, an accurate
refraction compensation scheme is needed. In this report, we
provide an algorithm that will compensate the error when the
object is at a finite range from the sensor. Sensitivities of
compensation errors to object range error, atmospheric model
mismatch, and possible statistical variations about a given
atmospheric model are also provided.
... iii
Ah "
CONTENTS
Abstract iii
I. INTRODUCTION 1
II. RAY TRACING IN SPHERICAL COORDINATES 5
2.1 Fermat's Principle 5
2.2 The Euler Equations in Spherical Coordinates 6
2.3 The Sphericaly Symmetric Case 8
III. ELEVATION REFRACTION ERROR FOR SENSOR VIEWING 9THROUGH ATMOSPHERE
IV. REFRACTIVE INDEX OF AIR 15
4.1 Model Atmospheres 15
4.2 Elevation Refraction Error Computation For A 20Given Atmospheric Model
4.3 A Statistical Model For Atmospheric 25Uncertainties
4.4 Geometric Considerations 31
V. SUMMARY AND CONCLUSIONS 35
ACKNOWLEDGMENTS 36
REFERENCES 37
v 1PREVIOUS PAGFTv IS BLANK
I. INTRODUCTION
The refraction of light by the earth's atmosphere
has been observed by astronomers for centuries. In order to
obtain the actual star location, one needs to know the path of
the light ray from the star through the atmosphere. Mathemat-
ical formulas that trace a light ray through nonuniform medium
have been derived and can be used for this purpose.[1]-[2]
With limited atmospheric data around the earth, it was impos-
sible to compensate the bending of light accurately. Until
recently, numerous data have been collected around the earth
in different seasons and at different altitudes. Atmospheric
models have been generated and used to correct these errors,
for example six atmosphere models are provided in a computer
code for atmospheric transmittence and radiance computation
called LOWTRAN 5.[31
A wide range of passive optical sensor applications
often require the sensor operate within the atmosphere while
the objects it examines are outside the atmosphere. When the
measurement accuracy at low elevation angle is important for
the sensor application, an accurate refraction compensation
scheme is needed. There are computer codes, for example
LOWTRAN 5[3], which can be used to calculate the refraction
error and compensate it. It is, however, not applicable to
the case when an object is at a finite range from the sensor.
1?
'I
Figure 1.1 shows that for each true target position, the
apparant elevation angle can be determined uniquely. However,
given the apparent elevation angle, the true target elevation
is determined only if the range to the target is known. This
is the purpose of this report to provide an algorithm that
will generate the actual refraction error for objects with
finite range and also provide some sensitivites of these
errors to the object range, the underlying atmospheric model
and possible statistical variations about a given atmospheric
model.
The report is organized in five sections. Section 2
provides a brief review of Fermat's principle and derives an
important formula for ray tracing in spherical coordinates.
In Section 3, a specific form of refractive index (a ratio of
velocity of light in vacuum and in a medium) is assumed,
namely, it is a spherically symmetric function about the cen-
ter of the earth. The formula that computes the refraction
error is derived for an object with a range R from the ob-
server. A simplified version of Edlen's expression for the
refractive index of air [41 is used in Section 4 to generate
the refractive index of air at different altitudes from at-
mospheric profiles provided by LOWTRAN 5. The sensitivities
of compensated elevation refraction errors to the object
range, atmospheric model mismatches and the statistical varia-
2
A L -
OPTICAL PATHVACUUM
+ Pw
PP
" DISTANCE OF P1 TO THECENTER OF EARTH
r0
r- - RADIUS OF THE EARTHATMOSPHERE
A- - REFRACTION ERROR, IF
OBJECT IS AT -
A - REFRACTION ERROR, IF
OBJECT IS AT RANGE R10,CENTER OF EARTH
Fiq. 1.1 Por a given apparent elevation angle, the trueelevation angle and hence the refraction error is a functionof the target range R.
3
tions about a given model atmosphere are presented. A surt,*"ry
and conclusion is given in Section 5.
4
II. RAY TRACING IN SPHERICAL COORDINATES
When light rays are propagated through an
atmospheric medium of continuously varying refractive index,
they experience a change in direction or refractive bending.
In this section, we will derive the equations for ray tracing
t in spherical coordinates from Fermat's principle.
2.1 Fermat's Principle[l]
Let n be the refractive index of a light ray in a
medium. The optical length of a ray which joins points P1 and
P2 (denoted by [P1P21) is given by
P 2[PP 2 ] = fp n ds (2.1)
where s is the distance measured along the light ray.
Fermat's principle states that a light ray always
chooses a trajectory that minimizes the optical length. In
mathematical terms, Fermat's principle assumes the form,
[PP 2 = f n ds = minimum. (2.2)
Instead of the optical length, we can introduce the concept of
transit time by dividing (2.2) by the constant c, the velocity
of light in vacuum. Since the velocity of light in a medium
with refractive index n is c/n, we have
5
P2(P 1P 2] c fP2 dt. (2.3)
Therefore, Fermat's principle is also known as the principle
of least time.
2.2 The Euler Equations in Spherical Coordinates
Let us consider that the refractive index is a
smooth and continuous function of position in spherical
coordinates and is denoted by n(r,e,fl. We use the definition
of the line element
ds = dr2 - /r2de2 + r 2 sin2 Od 2
= /1 + r2e' 2 + r2 sin 2 0012 dr (2.4)
with
- dr dr (2.5)
to express (2.1) in the form
f2 ,€,)dr = minimum. (2.6)
The function L is given by
, n(r,O,O)/1 + r2 0 2 + r 2 sin 2' (2.7)
In variational calculus, the function L in (2.6) is called the
6
Lagrangian. The solution of (2.6) is well known and can be
obtained by solving the Euler equations:
d aL ad I - = 0, (2.8)dr a8' 30
d aL aLdr a' - 0. (2.9)
Substituting (2.7) into (2.8) and (2.9), we have
d 2 - 9n nr 2sinecose (210)
dr s s' t- + s' "
2.2d nr sin = S - (2.11)
with
=ds
s > 1. (2.12)
With (2.12) and the chain rule, these equations can be written
as
d 2 do 2 an d2
d- ( n r an + nr 2sinec se(d-) (2.13)
d 2 .2 d#i and (nr sin 21) = 2 • (2.14)
These two equations should be sufficient to determine the ray
trajectory. The corresponding Euler equation for r can be
derived from (2.13) and (2.14) using the constraint (2.4).
7
2.3 The Spherically Symmetric Case
For the case where the refractive index depends only
upon r, (2.13) and (2.14) become
d (nr2 d) nr2sinecos8() 2.15)
d (nr2 sin ) = 0. (2.16)
Integrating (2.16), we have
2 .2e d =_ 2nr sin 2 ds (2.17)
We may choose the coordinate system so that 0 initially.
Then, C1 = 0 and (2.17) yields
d = 0
for all s. This reduces the problem into a two dimensional
problem, that is
d 2do(nr .) = 0. (2.18)
This result will be used in the following sections to
calculate the error of angular measurement due to refraction
in an atmosphere when spherical coordinates are specified and
the refractive index of the atmosphere is given.
AL8
III. ELEVATION REFRACTION ERROR FOR SENSOR VIEWINGTHROUGH ATMOSPHERE
In this section, we will limit ourselves to describ-
ing the gross characteristics of atmospheric refraction.
Other causes of refraction such as irregularities and varia-
tions within the earth's atmosphere which can not be deter-
mined by theory will not be considered. Furthermore, we
assume the refractive index of the earth's atmosphere is
spherically symmetric and depends only upon r, the distance to
the center of the earth. The non-sphericity of the earth can
be taken into account locally while applying results of this
section by using the effective earth radius at a given lati-
tude and the associated model atmosphere.
Let us consider a light ray passing through P 0 and
P1 in the earth's atmosphere as shown in Figure 3.1. Due to
the spherically symmetric assumption, the trajectory of the
light ray lies in a two dimensional plane and can be repre-
sented in polar coordinate as shown in Figure 3.1. Further-
more, the trajectory, s(r,O), satisfies (2.18), or equivalent-
ly,
2 donr T = constant. (3.1)
Let us define the ray inclination angle, *, to be
the angle between the tangential vector of the ray and the
9
OPTICAL PATH
ds7.00, r LOCAL HORIZON
dO ATMOSPHERE
EARTH "
ro r
CENTER OF EARTH
dOds -os I
Fig. 3.1 The trajectory of a light ray lies in a twodimensional plane.
10
local horizon. Then, (3.1) becomes
n r cos * = CO (3.2)
where C is determined from the initial values of n, r and
at P0 , that is
n r cos 4 = n0 r0 cos *0. (3.3)
If the refractive index were constant between P0 and P1, (for
example, the atmosphere removed), (3.3) reduces to
r 1 cos 41 = r0 cos 1P0" (3.4)
The equality holds when the ray is a straight line. The
angle, *0(=pi) is commonly called the elevation angle of an
object at P1 with respect to an observer at P0. The range, R,
is defined as the distance between P0 and P "
As demonstrated in Figure 3.2, an object at P1
(r=r ) with a range R from P would be observed to have an1 0
apparent elevation angle i0 at P0 , as if it were seen at PI*
with a range R from P0. If no atmosphere were present, the
true elevation angle would be a as indicated in Figure 3.2.
The elevation error due to refraction is defined as
Aa = 4'0 - a. (3.5)
Let 4I be the angle between the tangential vector of
the ray at P1 and the local horizon as shown in Figure 3.2.
Then, we have
S= 1- 8 (3.6)
ii
OPTICAL PATHVACUUM
I~ ~ PI
ATMOSPHERE
II EARTH
r0 - APPARENT ELEVATION
a - TRUE ELEVATION
f3 -TOTAL EARTH ANGLE BETWEENPO AND P,
CENTER OF EARTH
ELEVATION ERROR
=L = #10 - a
Fig. 3.2 The elevation error due to refraction when thetarqet is at a range R.
12ih
......
where
Cos (nr (3.7)
and
f 1 do , (3.8)r
with CO = n o r 0 cos 0 and no, n, are refractive indexes at r0and r,, respectively.
rUsing the fact that sin i _ = , (3.1) can be re-
written in the form
nr 2sinl d = CO (3.9)
Substituting (3.9) into (3.8) and sino by a function of cos*,
and using (3.2), we obtain
r C drB f- 1 0 (3.10)
0 nr 2 / C0/nr)
Notice that the refractive index n is a function of r. There
is no closed form expression of this integral in general.
Whenever the function n is specified, a can be determined by
numerical integration for a given *0.
For a given set of parameters (a,R,r0 ), Aa is deter-
13
A
mined by solving *0' such that
r 1sins - R cosa (3.11)
where 0 is a function of as given by (3.10) and
R ' 2 + r02+ 2Rr 0S inCe.
For the case when n n n0 over [r,,rl], we have
8 cos- (C 0/n 0 r,) -cos- (C0/n~r 0 ) (3.12)
or
8 - 1 (3.13)
It is easy to see that Aa=O and (3.11) is satisfied.
14
IV. REFRACTIVE INDEX OF AIR
As mentioned in Section 1, the refractive index, n,
of a medium for a light ray is defined as
n = v/c (4.1)
where c is the velocity of the light propagating in vacuum
while v is the velocity in the medium. The velocity of a
light ray propagating through the atmosphere varies with
changes in atmospheric composition, pressure, and tempera-
ture. It is strongly wavelength dependent at optical wave-
lengths, but it is not affected appreciably by water vapor.
In this report, we adopt a simplified version of Edlen's
expression for the refractive index of air[ 4I as follows:
0-6 45/2 Pn 1 + 10 x (77.46 + 0.459/x ) (4.2)
where = wavelength in micrometers (um)
P = atmospheric pressure in millibars
T = atmospheric temperature in deqrees Kelvin.
The atmospheric temperature and pressure variations
are principally functions of altitude, season and latitude.
4.1 Model Atmospheres
Six model atmospheres used in LOWTRAN 5(31 (a com-
puter code for atmospheric transmittance and radiance computa-
15
..
110 I I I I I I I- U.S. STANDARD (1962)
100- ....... MIDLATITUDE SUMMERMIDLATITUDE WINTER
90 -- SUBARTIC SUMMER.... SUBARTIC WINTER
80- TROPICAL
70
S60-
40
30
20
10
0180 200 220 240 260 280 300
TEMPERATURE (K)
Fiq. 4.1 The temperature profiles as a function of altitude.
16
IL
tion) are considered in this report. The model atmospheres
correspond to the 1962 U.S. standard atmosphere and five
supplementary models; that is, Tropical (15°N), Midlatitude
Summer (450 N, July), Midlatitutde Winter (45*N, January),
Subarctic Summer (600N, July ), and Subarctic Winter (600N,
January). The different models are digitized in 1-km steps
from 0 to 25 km, 5-km steps from 25 to 50 kin, then at 70 km
and 100 km directly as qiven by McClatchey et.al.,[5).
The temperature profiles for these models as a
function of altitude are shown in Figure 4.1. The pressure
profiles are given in Figure 4.2. Figure 4.3 shows the
profiles of pressure to temperature ratio for these models.
Notice that the 1962 U.S. standard model has a pressure to
temperature ratio profile very close to the mean of these
profiles. The refractive modulus
N = (n-1) x 106 (4.3)
which is proportional to the pressure-to-temperature ratio
according to Equation (4.2), is also given in Figure 4.3 at
S= 11UM.
In the next subsection, we will describe the scheme
of calculating the refraction error for an object at a given
range and elevation angle. Sensitivities of refraction errors
to the object range, atmospheric model mismatch will be
presented.
17
110
100-
-U.S. STANDARD (1962).................. MIDLATITUDE SUMMER
80 --- MIDLATITUDE WINTER80- --- SUBARTIC SUMMER
70 -SUBARTIC WINTER
E TROPICAL
w60-
t: 50-
40-
30-
20-
10-
0110-4 10-3 10-2 10-1 100 101 102 103 104
PRESSURE (mb)
Fiq. 4.2 The pressure profiles as a function of altitude.
18
70 \.
LOWTRAN 5 MODEL ATMOSPHERES60-.U.S. STANDARD (1962)
.......... MIDLATITUDE SUMMER
50 - MIDLATITUDE WINTER50 --
SUBARTIC SUMMERE....... SUBARTIC WINTER-
"' TROPICALr3 40D
< P PRESSURE (mb)
30 T TEMPERATURE (Kelvin)
n. INDEX OF REFRACTION1 +Nx 10-6
20 N EDLEN'S EXPRESSIONN (77.46 + 0.459/A 2 ) P/TA= 11/z
100.0002 0.001 0.01 0.1 10 20
P/T
0.02 0.1 1.0 10 100
Fig. 4.3 The profiles of pressure to temperature ratio as afunction of altitude.
19
4.2 Elevation Refraction Error Computation For A GivenAtmospheric Model
In this subsection, we apply the refraction equations
and various atmospheric models to show how well the true
target elevation can be estimated from the apparent target
elevation and approximate knowledge of the target range and
atmospheric parameters.
Let us assume that the refractive index of a given
atmospheric model for a light ray of wavelength X0 can be ex-
pressed by the following piecewise constant function
n(h) = n. for h i< h < h (4.4)nh nI i-i-- i1
where h is altitude in km and hI's are altitude steps defined
1
in the LOWTRAN 5 code. Substituting (4.4) into (3.10), we
have
-I -1
= cos [C0/nM+ (Re+hT) ] - cos [C0/nM+ 1 (Re+hM)]
M-1 1c s l= -1
-[C0/ni+1 ( R e + h i + 1 -Cos- cc0/ni+ (Re+hi)]1(4.5)i=i
+COS- [C 0/n L(R e+h L ) ] - 0
J.A.'2
where
C 0 n L(Re+hs)COS*0 (4.6)
Re is the earth radius and h s , hT are the sensor and target
altitudes, respectively. The indices L and M are determined
by hs and hT as below:
h < h < hL-1- s L
(4.7)
h M < hT < hM+ I
For a given set of parameters (a,R,h ), an iterative schemes
has been used to solve for *0 such that (3.11) is satisfied,
that is
(Re + h T)sin8 = R cosa (4.8)
where a is a function of ip0 as given in (4.8) and hT is ob-
tained by the following relation:
hT = [(R e+h s)2 + R2 + 2R(R e+h s)sincl 1/ 2 - R . (4.9)
As illustrated in Figure 4.4, the iterative scheme
used in this report is described as follows:
1. For a given set of parameters (a,R,h s), determine
hT, L and M according to Equations (4.9) and (4.7),
respectively. Let = a and j = 1.
21
0--PTICAL PATH
-~ 1st ITERATION
/32 GIVEN a h
CNEFEARTH A 1 =aa
roJ r 1 Aa0
fl22
pj ~ ~ ~ ~ ~ ~ ~ .fl(JhhT E.(.)
2. Compute aj by (4.5) for the given *0.
3. A new target range is computed as
R= [(Re+hs )2 + (Re+hT) 2 - 2(Re+h s)(Re+hT)cosl1 / 2
and the associated elevation angle as
-1 ~ iDRa. = cos [(Re+hT)sinaj/R
The difference of a and aj is
Aa. = a - a..) J
4. If
aIjJ < e (tolerance limit of the algorithm). (4.10)
go to step 5. Otherwise set
j = j + 1
0 = 0+
and go to step 2.
5. The elevation error due to refraction is computed as
A =0 - a
and the iteration procedure stops.
Figure 4.5 shows uncompensated elevation errors due
to refraction in V radians as a function of the actual object
elevation angle with different ranges (from 500 km to w) from
a sensor at an altitude of 15 km. The atmospheric model used
here is the 1962 U.S. standard atmosphere provided by LOWTRAN
5. In the case where the range information of the target is
23
1000
M R==
R =R 500 km
0nr
uJ 100z0_o
w-jw
-1 0
zw
0z SENSOR ALTITUDE: 15 km
U.S. STANDARD ATMOSPHERE (1962)
1 10 100
ELEVATION ANGLE (deg)
Fig. 4.5 The uncompensated elevation errors due toatmospheric refraction.
24
J . . .
not available, the elevation angle is compensated for a preset
range, say 1000 km in the example shown in Figure 4.6. Bias
errors for targets at ranges of 500, 1000 and - km are shown
in Figure 4.6 for the sensor presented in Figure 4.5. For
other atmospheric models, bias errors due to mismatched range
follow similar trends and magnitudes as those with the 1962
U.S. standard profile; they are shown in Figure 4.7. For a
mismatched atmospheric profile, Figure 4.8 shows bias errors
for objects at a common range of 1000 km for all atmospheric
models which are compensated by the 1962 U.S. standard profile
with the true range. The maximum refraction bias that
includes range and model mismatches is presented in Figure 4.9
with thereference model and range indicated. This figure
presents a spread of elevation biases over a wide range of
atmospheric mismatches. For actual applications, the
atmospheric profile is known to a certain accuracy.
Mismatches presented in Figure 4.9 will only occur in a few
extreme cases. A statistical model of the atmospheric profile
should be used to evaluate the elevation errors in a
statistical sense.
4.3 A Statistical Model For Atmospheric Uncertainties
It would be very complicated to express atmospheric
temperature and pressure variations with a statistical model.
To evaluate the elevation bias due to refraction, we only need
25
200 I t III I
COMPENSATED AT 1000 km RANGEWITH MATCHED ATMOSPHERIC
100
L <
-o R =1000 km
0
C
-100
-20I I I I I II I1 2 3 4 5 6 7 8 9 10 20 30 40 50
ELEVATION ANGLE (deg)
Fig. 4.6 The bias errors for targets at range indicated when
the elevation angle is compensated for a preset range.
26
200 -T
COMPENSATED AT 1000 km RANGE-WITH MATCHED ATMOSPHERIC
R=100 R PROFILE
C,,
- R =1000 km
z 0 -
0O -- L) _ -- U.S. STANDARD (1962) _
-- -- MIDLATITUDE WINTER
R 500 km--- SUBARCTIC SUMMER
.....- SUBARCTIC WINTER
TROPICAL
-200I I 12 3 4 6 7 8 910 20 30 40 50
ELEVATION ANGLE (deg)
Fig. 4.7 The bias errors due to mismatched range setting inthe compensation for different atmospheric models.
27
p
200 li
OBJECT AT 1000 km RANGE- COMPENSATED WITH THE 1962
100- U.S. STANDARD ATMOSPHERE
U,
ix-10 MIDLATITUDE SUMER
-SUBARCTIC SUMMER
SUBARCTIC WINTER
-TROPICAL
-200 1 I i1 2 3 4 5 67 89 10 20 30 40590
ELEVATION ANGLE (deg)
Fig. 4.8 The bias errors due to mismatched atmospheric modelin the compensation for a matched range setting.
28
SbokkALi
110
100 FOR APPARENT ELEVATION ANGLE WITH RANGE 500 km
COMPENSATED BY THE U.S. STANDARD PROFILE AT
90 1000 km RANGE
80 U.S. STANDARD (1962)
....... MIDLATITUDE SUMMERco 70
z .... MIDLATITUDE WINTER0 6 0 -
60 SUBARCTIC SUMMER
5SUBARCTIC WINTER- 50L.cc TROPICAL
40
230 '*
1: -
10 20 30 40 50 60
ELEVATION ANGLE (deg)
Fiq. 4.9 The maximum refraction bias that includes the range
and reference model mismatches.
29
. . .- S.
to know the pressure to temperature ratio. It is much
simpler to model the fluctuations of this ratio. As we have
seen in Figure 4.3, the variations of this ratio are very
small for a wide range of atmospheric models. As mentioned
previously, the 1962 U.S. standard atmosphere model has a
pressure to temperature ratio very close to the mean of the
six atmospheric models considered. In this report, we
construct a statistical model of the pressure to temperature
ratio for an atmosphere which has mean profile identical to
the 1962 U.S. Standard profile as follows:
(P/T)i = (P/T)i+ - (z i+-zi)(A z(P/T)i + a iui; 1i=,2,..,34
(P/T) = 0, (4.11)34
where
Az (P/T)i = (Pi+i/Ti+1 - Pi/Ti)/(zi+1-zi)
Pi = Atmospheric pressure in millibars of the 1962 U. S.1
Standard atmosphere at altitude zi
T. = Atmospheric temperature in degrees Kelvin of the1
1962 U.S. Standard atmosphere at altitude zi
a = Standard deviation of A z(P/T)i which is a half of
the sample standard deviation of the six atmospher-
ic models provided by LOWTRAN 5 at altitude zi
= A random normal distributed number with mean zero
and unit standard deviation.
30
A ..
Figure 4.10 and 4.11 show the mean and standard deviation of
50 Monte Carlo runs of elevation errors after compensation.
The same conditions are assumed here as those in Figures 4.6 -
4.9. It is interesting to note that the mean error is very
close to the bias presented in Fig. 4.6 for the deterministic
case The standard deviation is less than 200 Urad at 1 ele-
vation angle for the worst case.
4.4 Geometric Considerations
From the results presented in this section, it is
seen that refraction errors become much more significant at
low elevation angles. The elevation angle is a function of
target and sensor altitudes and the distance between them as
shown in Figure 4.12.
31
200~il~i
COMPENSATED AT 1000 km RANGEWITH THE 1962 U.S. STANDARD
_100
cn
z 00
-100-
-200 I li1 2 3 4 5 67 89 10 20 30 40 50
ELEVATION ANGLE (deg)
Fig. 4.10 The mean of elevation errors after compensation fora statistical atmosphere model.
32
200
R== -z0
-150150 COMPENSATED AT 1000 km RANGE
w R = 1000 km WITH THE 1962 U.S. STANDARDPROFILE
z 100
I-
0cr R 500km
z 500
U-w
cc 01 2 3 4 5 6 7 8 9 10 20 30 40 50
ELEVATION ANGLE (deg)
Fig. 4.11 The standard deviation of elevation errors aftercompensation for a statistical atmosphere model.
33
.
200 ELEVATION ANGLE (deg)20 10 5 43 2 1 0
_100
0 I I I100 200 400 800 1000 2000
RANGE (km)
Fig. 4.12 The elevation angle of target as a function oftarget altitude and range to a sensor at 15 km altitude.
34
V. SUMMARY AND CONCLUSIONS
A formula which can be used to compute the refrac-
tion error of an object at a finite range in a given atmos-
phere is derived. An algorithm which obtains an approximate
solution of the formula is presented. Sensitivities of the
bias error to the range and/or atmospheric model used in the
compensation scheme are also presented. A statistical model
which simulates the effects of the refraction errors due to
fluctuations in the atmospheric temperature and pressure is
presented. Simulation results are presented. From various
sensitivity studies presented here, it is important to
conclude that the most important portion of the bias error is
due to the range mismatch. The bias error can be minimized by
carefully selecting a reference range within the operational
range of the sensor.
A major conclusion from the results presented above
is that refraction errors are most serious at low elevation
angles.
35
ACKNOWLEDGMENTS
The author would like to thank Drs. Hsiao-Hua Burke
and Ronald P. Espinola for technical discussions, to Dr. S.
D. Weiner for reviewing the manuscript, to C. W. Edwards for
programming assistance, and to Chris Tisdale for preparing the
manuscript.
36
REFERENCES
1. M. Born and E. Wolf, Principles of Optics: Electromagnet-ic Theory of Propagation, Interference and Diffraction ofLight, 6th Edition (Pergamon Press, Oxford, England,1983).
2. D. Marcuse, Light Transmission Optics (Van NostrandReinhold, New York, 1972).
3. F. X. Kneizys et al., "Atmospheric Transmittance/Radiance: Computer Code LOWTRAN 5," AFGL-TR-80-0067,Environmental Research Papers, No. 697, Air ForceGeophysics Laboratory, Hanscom APB (21 February 1980).
4. B. Edlen, "The Refractive Index of Air," Metrologia 2,71 (1966).
5. R. A. McClatchey et al., Optical Properties of theAtmosphere, Third--Ed-Eion, AFCRL-72-0497, AD753 075(24 August 1972).
37
AL A&L4
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READ) INSTRUCTIONSREPORT DOCUMENTATION PAGE [IEFORE COMiPLT ING; FORM1. REPORT NUBR2 GO ~ ACSINO. 3. RECIPIENT'S CATALOG NUMBER
ESD-TR-84-015 -I A N iv 37 74. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED
Atmospheric Refraction Error and Its Compensation Technical Reportfor Passive Optical Sensors S. PERFORMING ORG. REPORT NUMBER
Technical Report 6867. AUTHOR(s) 1. CONTRACT OR GRANT NUMBER(@)
Keh.Ping Dunn F19628-80.( -0002
S. PERFORMING ORGARIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKLincoln Laboratory, M.I.T. AE OKUI UBRP.O. Box 73 Program Element No. 63304ALexington, MA 02173-0073
11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATEBallistic Missile Defense Program Office 4 Jun 1984Department of the Armsk5001 Eisenhower Avenue 13. NUMBER OF PAGESAlexandria. VA 22333 46
III. MONITORING AGENCY NAME &ADDRESS (if diffennfrom Controlling Office) IS. SECURITY CLASS. (of this report)
Electronic Systems Div ision UcasfeHanscom AFB, MA 01731 Ino. DECLASSIFICATION DOWNGRADING SCHEDULE
1S. DISTRIBUTION STATEMENT (of this Report)
Approved for public release. distribution unlimited.
17. DISTRIBUTION STATEMENT (of the aubstrat entered in Block 20, if different fin Report)
1B. SUPPLEMENTARY NOTES
None
19. KEY WORDS (Continue on nreere side if necessary awl identify by block number)
passive optical sensors range errorlow elevation angle atmospheric model mismatchlight refraction compensation statistical variationsalgorithm apparent elevation angle
29. ABSTRACT (Continue on reeree side if necessary and identify by block nunmber)
A wide range of passive optical sensor applications often require the sensor to operate within theatmosphere while the objects it examines are outside the atmosphere. The refraction of light by theearth's atmosphere becomes a significant error source when the objects are at low elevation angle. Whenthe measurement accuracy at low elevation angle is important for the sensor application, an accurate re-fraction compensation scheme is needed. In this report. we provide an algorithm that will compensatethe error when the object is at a finite range from the sensor. Sensitivities of compensation errors to ob-ject range error, atmospheric model mismatch, and possible statistical variations about a given atmo-spheric model are also provided.
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